logarithm...

In mathematics, the logarithm of a number to a given base is the power or exponent to which the base must be raised in order to produce the number.
For example, the logarithm of 1000 to the base 10 is 3, because 10 raised to the power of 3 is 1000; the base 2 logarithm of 32 is 5 because 2 to the power 5 is 32.
The logarithm of x to the base b is written logb(x) or, if the base is implicit, as log(x). So, for a number x, a base b and an exponent y,

An important feature of logarithms is that they reduce multiplication to addition, by the formula:

That is, the logarithm of the product of two numbers is the sum of the logarithms of those numbers. The use of logarithms to facilitate complex calculations was a significant motivation in their original development.
When x and b are restricted to positive real numbers, logb(x) is a unique real number. The magnitude of the base b must be neither 0 nor 1; the base used is typically 10, e, or 2. Logarithms are defined for real numbers and for complex numbers. [1][2]
The major property of logarithms is that they map multiplication to addition. This ability stems from the following identity:

which by taking logarithms becomes

For example,



A related property is reduction of exponentiation to multiplication. Using the identity:

it follows that c to the power p (exponentiation) is:

or, taking logarithms:

In words, to raise a number to a power p, find the logarithm of the number and multiply it by p. The exponentiated value is then the inverse logarithm of this product; that is, number to power = bproduct.
For example,

Besides reducing multiplication operations to addition, and exponentiation to multiplication, logarithms reduce division to subtraction, and roots to division. For example,


Logarithms make lengthy numerical operations easier to perform. The whole process is made easy by using tables of logarithms, or a slide rule, antiquated now that calculators are available. Although the above practical advantages are not important for numerical work today, they are used in graphical analysis.

The logarithm as a function
Though logarithms have been traditionally thought of as arithmetic sequences of numbers corresponding to geometric sequences of other (positive real) numbers, as in the 1797 Britannica definition, they are also the result of applying an analytic function. The function can therefore be meaningfully extended to complex numbers.
The function logb(x) depends on both b and x, but the term logarithm function (or logarithmic function) in standard usage refers to a function of the form logb(x) in which the base b is fixed and so the only argument is x. Thus there is one logarithm function for each value of the base b (which must be positive and must differ from 1). Viewed in this way, the base-b logarithm function is the inverse function of the exponential function bx. The word "logarithm" is often used to refer to a logarithm function itself as well as to particular values of this function.
The base can also be a complex number; the evaluation of the log is just slightly more complicated in this case. See imaginary base.

[edit] Logarithm of a complex number
When the base b is real and z is a complex number, say z = x + i y, the logarithm of z is found by putting z in polar form that is, z = (x2 + y2)1/2 exp (i tan−1 (y / x) ). If the base of the logarithm is chosen as e [3], that is, using loge (denoted by ln and called the natural logarithm), the logarithm becomes:



This evaluation uses the properties of all logarithms (see above), regardless of choice of base: logb (c d ) = logb (c ) + logb (d ) and its generalization to arbitrary products logb bz = z. Because the inverse tangent is a multiple valued function of its argument, the logarithm of a complex number is not unique either. See article on complex logarithm.

Group theory
From the pure mathematical perspective, the identity

is fundamental in two senses. First, the remaining three arithmetic properties can be derived from it. Furthermore, it expresses an isomorphism between the multiplicative group of the positive real numbers and the additive group of all the reals.
Logarithmic functions are the only continuous isomorphisms from the multiplicative group of positive real numbers to the additive group of real numbers.

Bases
The most widely used bases for logarithms are 10, the mathematical constant e ≈ 2.71828... and 2. When "log" is written without a base (b missing from logb), the intent can usually be determined from context:
natural logarithm (loge, ln, log, or Ln) in mathematical analysis, statistics, economics and some engineering fields. The reasons to consider e the natural base for logarithms, though perhaps not obvious, are numerous and compelling.
common logarithm (log10 or simply log; sometimes lg) in various engineering fields, especially for power levels and power ratios, such as acoustical sound pressure, and in logarithm tables to be used to simplify hand calculations
binary logarithm (log2; sometimes lg, lb, or ld), in computer science and information theory
indefinite logarithm (Log or [log ] or simply log) when the base is irrelevant, e.g. in complexity theory when describing the asymptotic behavior of algorithms in big O notation.
To avoid confusion, it is best to specify the base if there is any chance of misinterpretation.

Other notations
The notation "ln(x)" invariably means loge(x), i.e., the natural logarithm of x, but the implied base for "log(x)" varies by discipline:
Mathematicians understand "log(x)" to mean loge(x). Calculus textbooks will occasionally write "lg(x)" to represent "log10(x)".
Many engineers, biologists, astronomers, and some others write only "ln(x)" or "loge(x)" when they mean the natural logarithm of x, and take "log(x)" to mean log10(x) or, in computer science, log2(x).
On most calculators, the LOG button is log10(x) and LN is loge(x).
In most commonly used computer programming languages, including C, C++, Java, Fortran, Ruby, and BASIC, the "log" function returns the natural logarithm. The base-10 function, if it is available, is generally "log10."
Some people use Log(x) (capital L) to mean log10(x), and use log(x) with a lowercase l to mean loge(x).
The notation Log(x) is also used by mathematicians to denote the principal branch of the (natural) logarithm function.
In some European countries, a frequently used notation is blog(x) instead of logb(x).[4]
This chaos, historically, originates from the fact that the natural logarithm has nice mathematical properties (such as its derivative being 1/x, and having a simple definition), while the base 10 logarithms, or decimal logarithms, were more convenient for speeding calculations (back when they were used for that purpose). Thus natural logarithms were only extensively used in fields like calculus while decimal logarithms were widely used elsewhere.
As recently as 1984, Paul Halmos in his "automathography" I Want to Be a Mathematician heaped contempt on what he considered the childish "ln" notation, which he said no mathematician had ever used. The notation was in fact invented in 1893 by Irving Stringham, professor of mathematics at Berkeley.[citation needed]
In computer science, the base 2 logarithm is sometimes written as lg(x), as suggested by Edward Reingold and popularized by Donald Knuth. However, lg(x) is also sometimes used for the common log, and lb(x) for the binary log.[5] In Russian literature, the notation lg(x) is also generally used for the base 10 logarithm.[6] In German, lg(x) also denotes the base 10 logarithm, while sometimes ld(x) or lb(x) is used for the base 2 logarithm.
The clear advice of the United States Department of Commerce National Institute of Standards and Technology is to follow the ISO standard Mathematical signs and symbols for use in physical sciences and technology, ISO 31-11:1992, which suggests these notations:[7]
The notation "ln(x)" means loge(x);
The notation "lg(x)" means log10(x);
The notation "lb(x)" means log2(x).
As the difference between logarithms to different bases is one of scale, it is possible to consider all logarithm functions to be the same, merely giving the answer in different units, such as dB, neper, bits, decades, etc.; see the section Science and engineering below. Logarithms to a base less than 1 have a negative scale, or a flip about the x axis, relative to logarithms of base greater than 1.

Change of base
While there are several useful identities, the most important for calculator use lets one find logarithms with bases other than those built into the calculator (usually loge and log10). To find a logarithm with base b, using any other base k:

Moreover, this result implies that all logarithm functions (whatever the base) are similar to each other. So to calculate the log with base 2 of the number 16 with a calculator:


Uses of logarithms
Logarithms are useful in solving equations in which exponents are unknown. They have simple derivatives, so they are often used in the solution of integrals. The logarithm is one of three closely related functions. In the equation bn = x, b can be determined with radicals, n with logarithms, and x with exponentials. See logarithmic identities for several rules governing the logarithm functions.

Science
Various quantities in science are expressed as logarithms of other quantities; see logarithmic scale for an explanation and a more complete list.
In chemistry, the negative of the base-10 logarithm of the activity of hydronium ions (H3O+, the form H+ takes in water) is the measure known as pH. The activity of hydronium ions in neutral water is 10−7 mol/L at 25 °C, hence a pH of 7.
The bel (symbol B) is a unit of measure which is the base-10 logarithm of ratios, such as power levels and voltage levels. It is mostly used in telecommunication, electronics, and acoustics. The Bel is named after telecommunications pioneer Alexander Graham Bell. The decibel (dB), equal to 0.1 bel, is more commonly used. The neper is a similar unit which uses the natural logarithm of a ratio.
The Richter scale measures earthquake intensity on a base-10 logarithmic scale.
In spectrometry and optics, the absorbance unit used to measure optical density is equivalent to −1 B.
In astronomy, the apparent magnitude measures the brightness of stars logarithmically, since the eye also responds logarithmically to brightness.
In psychophysics, the Weber–Fechner law proposes a logarithmic relationship between stimulus and sensation.
In computer science, logarithms often appear in bounds for computational complexity. For example, to sort N items using comparison can require time proportional to the product N × log N. Similarly, base-2 logarithms are used to express the amount of storage space or memory required for a binary representation of a number—with k bits (each a 0 or a 1) one can represent 2k distinct values, so any natural number N can be represented in no more than (log2 N) + 1 bits.
Similarly, in information theory logarithms are used as a measure of quantity of information. If a message recipient may expect any one of N possible messages with equal likelihood, then the amount of information conveyed by any one such message is quantified as log2 N bits.
In geometry the logarithm is used to form the metric for the half-plane model of hyperbolic geometry.
Many types of engineering and scientific data are typically graphed on log-log or semilog axes, in order to most clearly show the form of the data.
In inferential statistics, the logarithm of the data in a dataset can be used for parametric statistical testing if the original data does not meet the assumption of normality.
Musical intervals are measured logarithmically as semitones. The interval between two notes in semitones is the base-21/12 logarithm of the frequency ratio (or equivalently, 12 times the base-2 logarithm). Fractional semitones are used for non-equal temperaments. Especially to measure deviations from the equal tempered scale, intervals are also expressed in cents (hundredths of an equally-tempered semitone). The interval between two notes in cents is the base-21/1200 logarithm of the frequency ratio (or 1200 times the base-2 logarithm). In MIDI, notes are numbered on the semitone scale (logarithmic absolute nominal pitch with middle C at 60). For microtuning to other tuning systems, a logarithmic scale is defined filling in the ranges between the semitones of the equal tempered scale in a compatible way. This scale corresponds to the note numbers for whole semitones. (see microtuning in MIDI).

Exponential functions
One way of defining the exponential function ex, also written as exp(x), is as the inverse of the natural logarithm. It is positive for every real argument x.
The operation of "raising b to a power p" for positive arguments b and all real exponents p is defined by

The antilogarithm function is another name for the inverse of the logarithmic function. It is written antilogb(n) and means the same as bn.

Easier computations
Logarithms can be used to replace difficult operations on numbers by easier operations on their logs (in any base), as the following table summarizes. In the table, upper-case variables represent logs of corresponding lower-case variables:
Operation with numbers
Operation with exponents
Logarithmic identity
These arithmetic properties of logarithms make such calculations much faster. The use of logarithms was an essential skill until electronic computers and calculators became available. Indeed the discovery of logarithms, just before Newton's era, had an impact in the scientific world that can be compared with that of the advent of computers in the 20th century because it made feasible many calculations that had previously been too laborious.
As an example, to approximate the product of two numbers one can look up their logarithms in a table, add them, and, using the table again, proceed from that sum to its antilogarithm, which is the desired product. The precision of the approximation can be increased by interpolating between table entries. For manual calculations that demand any appreciable precision, this process, requiring three lookups and a sum, is much faster than performing the multiplication. To achieve seven decimal places of accuracy requires a table that fills a single large volume; a table for nine-decimal accuracy occupies a few shelves. Similarly, to approximate a power cd one can look up log c in the table, look up the log of that, and add to it the log of d; roots can be approximated in much the same way.

The C and D scales on this slide rule are marked off at positions corresponding to the logarithms of the numbers shown. By mechanically adding the logs of 1.3 and 2, the cursor shows the product is 2.6.
One key application of these techniques was celestial navigation. Once the invention of the chronometer made possible the accurate measurement of longitude at sea, mariners had everything necessary to reduce their navigational computations to mere additions. A five-digit table of logarithms and a table of the logarithms of trigonometric functions sufficed for most purposes, and those tables could fit in a small book. Another critical application with even broader impact was the slide rule, an essential calculating tool for engineers. Many of the powerful capabilities of the slide rule derive from a clever but simple design that relies on the arithmetic properties of logarithms. The slide rule allows computation much faster still than the techniques based on tables, but provides much less precision, although slide rule operations can be chained to calculate answers to any arbitrary precision.

Related operations Cologarithms
The cologarithm of a number is the logarithm of the reciprocal of the number, meaning cologb(x) = logb(1/x) = −logb(x).[8]

Antilogarithms
The antilogarithm is the logarithmic inverse of the logarithm, meaning that the antilogb(logb(x)) = x. Thus, setting by = x implies that logb(x) = y. By taking the antilogb of both sides, antilogb(logb(x)) = antilogby, thus x = antilogby. Therefore, by = antilogby.

Calculus
The natural logarithm of a positive number x can be defined as

The derivative of the natural logarithm function is

By applying the change-of-base rule, the derivative for other bases is

The antiderivative of the natural logarithm ln(x) is

and so the antiderivative of the logarithm for other bases is

See also: Table of limits, list of integrals of logarithmic functions.

[edit] Series for calculating the natural logarithm
There are several series for calculating natural logarithms.[9] The simplest, though inefficient, is:
when
To derive this series, start with ()

Integrate both sides to obtain


Letting and thus , we get

A more efficient series is

for z with positive real part.
To derive this series, we begin by substituting −x for x and get

Subtracting, we get

Letting and thus , we get

For example, applying this series to

we get

and thus



where we factored 1/10 out of the sum in the first line.
For any other base b, we use

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